where \(k_p(q) \in \mathbf {N}\)  is the number of distance minimizing geodesics from \(p\)  to \(q\) . Since the set \(Q(p)\)  is closed in \(M\) , the complement \({\text{cut}}(p)\setminus Q(p)\)  is relatively open in \({\text{cut}}(p)\) , and there exists a open neighborhood \(W \subset M\)  of \(q\)  such that
\(Q(p)\cap W=\emptyset .\) 
Thus by the previous discussion for any \(x \in {\text{cut}}(p)\cap W\)  there are only \(k_p(x) \in \mathbf {N}\)  many distance minimizing geodesics connecting \(p\)  to \(x\) .
The following lemma is an adaptation of the analogous result given in [1]}.
